ECE 210 Review Session

MIDTERM TWO

ALEX ZHANG, JASON FLANAGAN, GRANT MCKECHNIE

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The Path

1. Circuits!
2. Generalize circuits into arbitrary linear systems
3. Analyzing linear systems by hand is painful (so much algebra, time derivatives…)
1. Three capacitors in a circuit = 3rd order differential equation. YUCK
4. Introducing the frequency domain!
• This is a pivotal moment. You will live and breathe in this domain for at least the remainder of your time here. The Fourier domain pops up in signal processing, power transfer, energy, communications, matrix/vector multiplication, antenna arrays, and so much more.
• So powerful that even quantum computing still uses it.
• The only 3-of-5 for EEs that doesn’t feature the Fourier domain is ECE391. Boring class anyway.
5. Phasors - single sinusoid, steady state only
6. Fourier series: Periodic functions, steady state only
7. Fourier Transform: Almost all functions, steady state only
8. Laplace Transform: Almost all functions, transient and steady state solutions

RL, RC, RLC Circuit

• ▪ RL and RC circuits require setting up and solving a first-order ODE
• ▪ RLC circuits require setting up and solving a second-order

At the steady state,

• Capacitors act as open circuits
• Inductors act as wires
• Continuous functions of time:
• Voltage across a capacitor
• Current through an inductor

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Second Order Differential Equations

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First and Second Order Circuits

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y(t) expressions

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Conceptual Questions (30 seconds each)

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A system converts its input f(t) = 64.27cos(2t+3000pi) + 20.3cos(4t) into a steady-state output

y(t) = 1038sin(2t – 1.25pi). Is the system LTI? Why?

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Available Power

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Resonance

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Filters

(1)

(2)

(3)

Steps to Find a Filter

• Remember, scale your input
• So response is output / input
• In order to determine type of filter, evaluate |H(w)| at 0, ∞, and in between (possibly at w0)

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High Pass

|H(0)|=0

|H(∞)|=1

Low Pass

|H(0)|=1

|H(∞)|=0

Band Pass

|H(0)|=0 |H(w0)|>0

|H(∞)|=0

Conceptual Questions

A linear system has the following frequency response:

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Find the output of the system when the input is

Conceptual Questions

A linear system is described by the following differential equation:

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Find the frequency response H(w) of the system.

Conceptual Question Solution

Conceptual Question Solution

Phasor Definition

Phasors, Co-sinusoids, and Impedance

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Once you’ve converted every circuit element in the phasor domain, you can analyze the circuit using all the ways covered in the first midterm!!!

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• Node Voltage
• Superposition
• Loop Current
• Source Transformations

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NOTE: The solution obtained after analysis with the phasor method is the STEADY STATE SOLUTION !!!

Formulae to Remember:

Conceptual Questions (30 seconds each)

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Conceptual Question

Convert the following circuit into the phasor domain:

Fourier Series

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Other properties of Fourier Series

• Division by 0 IS BAD. If a coefficient depends on division by 0, find it using another method.
• For discontinuities in the sequence: the Fourier series converges to the middle of the discontinuity.
• Gibbs phenomenon: there are ripples that appear right before a big jump, which we can only get rid of if we let n go to infinity.

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• Conditions for existence and usage of Fourier Series:
• - f(t) must be periodic (nonperiodic signals are dealt with using a Fourier Transform, you’ll learn this very soon ;)
• - f(t) must be absolutely integrable.
• - There must be a finite number of maxes and mins over one period, and a finite number of finite discontinuities over one period.

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Parseval’s Theorem

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Fourier Series:

Find the Exponential Fourier series of the following expression:

Fourier Series:

Find the Exponential Fourier series of the following expression:

Old HW Question

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Monster Problem

Find the exponential Fourier series of the following signal:

Monster Problem Solution

Step 1: Find T and w_0.

Monster Problem Solution

Step 2: Set up the integral

Monster Problem Solution

Step 3: Evaluate said integral

Monster Problem Solution

Step 3: Evaluate said integral

Can we simplify?

Monster Problem Solution

Step 3: Evaluate said integral

Monster Problem Solution

Step 4: Done!

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Or are we…

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Monster Problem Solution

What if n = 0?

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Yikes. We need a different equation.

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Monster Problem Solution

Step 2: Set up the integral, but now n = 0

Monster Problem Solution

Step 2: Evaluate the integral, but now n = 0

Monster Problem Solution

Now we’re done!

Monster Problem Solution

Now we’re done!

Monster Problem Part 2

Now we pass f(t) through a linear system with the following frequency response:

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What is the output y(t)?

Monster Problem Part 2

What is the output y(t)?

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Step 1: What actually are the frequencies in f(t)?

Monster Problem Part 2

What is the output y(t)?

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Step 1: What actually are the frequencies in f(t)?

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Answer: w = all integers!

Monster Problem Part 2

What is the output y(t)?

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Step 2: What frequencies will survive the filter?

Monster Problem Part 2

What is the output y(t)?

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Step 2: What frequencies will survive the filter?

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Answer: -3.7 to -3.3, -1.5 to 1.5, 2.5 to 3.5.

(Note that these are just guesses)

Monster Problem Part 2

What is the output y(t)?

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Step 3: What values of n survive the filter?

Monster Problem Part 2

What is the output y(t)?

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Step 3: What values of n survive the filter?

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n = -1, 0, 1, 3. Everything else is multiplied by 0 and thus goes away!

Monster Problem Part 2

What is the output y(t)?

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Step 4: Now calculate Y = HF only at n = -1, 0, 1, and 3.

Monster Problem Part 2

Step 4: Now calculate Y = HF only at n = -1, 0, 1, and 3.

Monster Problem Part 2

Step 4: Now calculate Y = HF only at n = -1, 0, 1, and 3.

Monster Problem Part 2

Step 4: Now calculate Y = HF only at n = -1, 0, 1, and 3.

Monster Problem Part 2

All together now…

Monster Problem Part 3

Calculate the average power of y(t).

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(Write down y(t), then I’ll give you the tables on the next slide)

Monster Problem Part 3

Average power: In the tables!

Monster Problem Part 4

Find the trigonometric and compact forms of y(t).

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(Write down y(t), then I’ll give you the tables on the next slide)

Monster Problem Part 4

Trigonometric form… What was the formula again?

Monster Problem Part 4

Ok. Let’s go!

Monster Problem Part 4

Altogether now…

Monster Problem Part 4

Compact form… What was the formula again?

Monster Problem Part 4

Ok. Let’s go!

Monster Problem Part 4

But wait… What’s the full formula?

Oh no. y(t) isn’t completely real.

COMPACT FORM FOR y(t) DOESN’T EXIST!!!!

Monster Problem Part 5

Ha I’m kidding.

Give yourself a pat on the back.

That was a hard one.

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Spring 2014 Question 1

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Spring 2018 Question 4

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Spring 2018 Question 4 Continued

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Fall 2017 Question 2

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Spring 2016 Question 5

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